Abstract: Using Chebyshev polynomials combined with some mild combinatorics, we providea new formula for the analytical planar limit of a random matrix model with aone-cut potential $V$. For potentials $Vx=x^{2}-2-\sum {n\ge1}a {n}x^{n}-n$,as a power series in all $a {n}$, the formal Taylor expansion of the analyticplanar limit is exactly the formal planar limit. In the case $V$ is analytic ininfinitely many variables $\{a {n}\} {n\ge1}$ on the appropriate spaces, theplanar limit is also an analytic function in infinitely many variables and wegive quantitative versions of where this is defined.Particularly useful in enumerative combinatorics are the gradings of $V$,$V {t}x=x^{2}-2-\sum {n\ge1}a {n}t^{n-2}x^{n}-n$ and$V {t}x=x^{2}-2-\sum {n\ge3}a {n}t^{n-2 -1}x^{n}-n$. The associated planarlimits $Ft$ as functions of $t$ count planar diagram sorted by the number ofedges respectively faces. We point out a method of computing the asymptotic ofthe coefficients of $Ft$ using the combination of the \emph{wzb} method andthe resolution of singularies. This is illustrated in several computationsrevolving around the important extreme potential$V {t}x=x^{2}-2+\log1-\sqrt{t}x$ and its variants. This particular examplegives a quantitive and sharp answer to a conjecture of t-Hoofts which statesthat if the potential is analytic, the planar limit is also analytic.